Lecture 18: Single-view modeling, Part 2

CS4670 / 5670 : Computer VisionNoah Snavely

Announcements

• Project 3 due this Thursday by 11:59pm– Demos on Friday (?)

• Project 3 artifacts due this Friday by 11:59pm

• Take-home prelim out Friday, due Wednesday 10/24 by the beginning of class– Please don’t post on Piazza about the prelim, just

email the staff

Quiz from last time

• How do you compute the image-space line l passing through two points p and q?

• How do you compute the image-space point p at the intersection of two lines l and m?

Comparing heightsVanishingVanishing

PointPoint

Measuring height

1

2

3

4

55.4

2.8

3.3

Camera heightHow high is the camera?

Measuring height without a ruler

C

Measuring height without a ruler

ground plane

Compute Z from image measurements• Need more than vanishing points to do this

Z

The cross ratio• A Projective Invariant– Something that does not change under projective

transformations (including perspective projection)

P1

P2

P3

P4

1423

2413

PPPP

PPPP

The cross-ratio of 4 collinear points

Can permute the point ordering• 4! = 24 different orders (but only 6 distinct values)

This is the fundamental invariant of projective geometry

1i

i

i

i Z

Y

X

P

3421

2431

PPPP

PPPP

vZ

r

t

b

tvbr

rvbt

Z

Z

image cross ratio

Measuring height

B (bottom of object)

T (top of object)

R (reference point)

ground plane

HC

TBR

RBT

scene cross ratio

1

Z

Y

X

P

1

y

x

pscene points represented as image points as

R

H

R

H

R

Measuring height

RH

vz

r

b

t

R

H

Z

Z

tvbr

rvbt

image cross ratio

H

b0

t0

vvx vy

vanishing line (horizon)

vz

r

b

t0

vx vy

vanishing line (horizon)

v

t0

m0

What if the point on the ground plane b0 is not known?• Here the guy is standing on the box, height of box is known• Use one side of the box to help find b0 as shown above

b0

t1

b1

Measuring height

3D Modeling from a photograph

St. Jerome in his Study, H. Steenwick

3D Modeling from a photograph

3D Modeling from a photograph

Flagellation, Piero della Francesca

3D Modeling from a photograph

video by Antonio Criminisi

3D Modeling from a photograph